Define power. Prove that P = F.v, where the symbols have their usual meanings.

In physics, power is the rate at which work is done or the rate at which energy is transferred or converted. It measures how quickly a certain amount of energy is used or produced per unit of time. Power is a scalar quantity and is typically measured in watts (W) in the International System of Units (SI).

Now, let's prove that power (P) can be expressed as the dot product of force (F) and velocity (v), assuming that the force is constant and applied in the direction of the velocity.

First, let's consider the definition of work (W) done by a force on an object:

Work (W) = Force (F) * Displacement (d) * cos(theta),

where theta is the angle between the force vector and the displacement vector.

If the force is applied in the direction of the displacement, then cos(theta) = 1, and the equation simplifies to:

Work (W) = Force (F) * Displacement (d).

Now, power (P) is defined as the rate of doing work, which means how much work is done per unit of time (t):

Power (P) = Work (W) / time (t).

Now, substituting the expression for work (W) in terms of force (F) and displacement (d):

Power (P) = (Force (F) * Displacement (d)) / time (t).

Since velocity (v) is defined as displacement (d) per unit of time (t):

v = d / t.

Rearranging the above equation, we get:

d = v * t.

Now, substituting this value of displacement (d) in terms of velocity (v) and time (t) in the power equation:

Power (P) = (Force (F) * (v * t)) / t.

The 't' in the numerator and denominator cancels out, and we get:

Power (P) = Force (F) * v.

Thus, we have proved that power (P) can be expressed as the dot product of force (F) and velocity (v):

P = F * v.

This result is valid when the force is constant and applied in the direction of the velocity. In more general cases, where the force and velocity vectors might have different directions, the power can still be calculated using the dot product of force and velocity vectors, taking into account the angle between them.